Question

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow \infty} \frac{\tan ^{-1} x}{x}$$

Short answer

Answer: The limit as \(x\) approaches infinity of the function \(\frac{\tan^{-1}(x)}{x}\) is 0.

Step-by-step solution

Differentiate the numerator and denominator

First, we need to find the derivative of the numerator and the derivative of the denominator: For numerator \(f(x) = \tan^{-1}(x)\): $$f'(x) = \frac{1}{1+x^2}$$ Reason: The derivative of \(\tan^{-1}(x)\) is \(\frac{1}{1+x^2}\) using the chain rule. For denominator \(g(x) = x\): $$g'(x) = 1$$ Reason: The derivative of \(x\) is \(1\).

Apply L'Hôpital's Rule

Now we will apply L'Hôpital's Rule: $$\lim _{x \rightarrow \infty} \frac{\tan^{-1}(x)}{x} = \lim_{x \rightarrow \infty} \frac{f'(x)}{g'(x)} = \lim_{x \rightarrow \infty} \frac{\frac{1}{1+x^2}}{1}$$

Evaluate the limit

Now we can evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{\frac{1}{1+x^2}}{1} = \lim_{x \rightarrow \infty} \frac{1}{1+x^2}$$ As \(x\) approaches infinity, the term \(x^2\) grows larger and dominates the denominator, so the entire fraction approaches zero. Therefore, the limit is: $$\lim _{x \rightarrow \infty} \frac{\tan ^{-1} x}{x} = 0$$

Key concepts

L'Hôpital's Rule

When you're faced with an indeterminate form like \(0/0\) or \(\infty/\infty\), L'Hôpital's Rule is the mathematical equivalent of a superhero swooping in to save the day. This powerful tool allows us to find the limit of a ratio of two functions by comparing the limits of their derivatives instead.

L'Hôpital's Rule states that if you have \(\lim_{x \to a} f(x)/g(x)\) and both \(f(x)\) and \(g(x)\) approach 0 or both approach infinity as \(x\) approaches \(a\), then the limit of the ratio is equal to the limit of the ratio of their derivatives, provided the limit exists. That's why our first step in the exercise was to find the derivatives of the numerator and the denominator.

Applying L'Hôpital's Rule

Once the derivatives are known, you simply take the limit of that new ratio. If the resulting limit is still an indeterminate form, don't fret! L'Hôpital's allows for multiple applications. Keep differentiating and substituting until you reach a determinate form or until it's clear the limit does not exist. Remember, L'Hôpital's Rule is available only when certain conditions are met; not all ratios of functions qualify for its application.

Infinite Limits

When we talk about infinite limits, we're dealing with scenarios where a function's value grows without bound as the input approaches a certain point, or extends towards infinity. It can seem a bit abstract, but it's an important concept in calculus to describe behavior at the edges of our mathematical world.

Here's what's fascinating: infinite limits aren’t confined to skyscraping values. They can also plunge deep into the negatives, teasing the abyss of negative infinity. Moreover, not all limits reaching towards infinity will yield an infinite limit; some might surprise you and settle down to a finite value as seen in our exercise with the arctangent function.

Identifying Infinite Limits

A limit is considered infinite if, as the variable \(x\) approaches a certain value or infinity, the function either increases or decreases without bounding to a certain number. The process of evaluation is much like standard limit calculations, but one's focus should be on the dominating terms, as they dictate the function's behavior at large values of \(x\).

Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions toss a fascinating twist into calculus. While you might be accustomed to the derivatives of basic trigonometric functions, their inverse counterparts have their own unique set of rules to play by.

The crux of understanding the derivative of inverse trigonometric functions lies in the relationship they share with their respective functions. They essentially undo what the original trigonometric function does and hence, their derivatives reflect that undoing process in a precise mathematical form.

Common Derivatives of Inverse Trigonometric Functions

For instance, the derivative of \(\tan^{-1} x\), as used in our textbook example, is \(1/(1+x^2)\). These formulas are derived using implicit differentiation, often involving a bit of trigonometric identity magic and algebraic manipulation. Mastering these derivatives equips you with the tools to tackle problems involving rates of change when the subject is an angle commonly described by its trigonometric function.